When I lived in Germany, I used a local (private) practice for my immediate needs. This was covered by my statutory (yet still private) insurance from a Krankenkasse. I usually could get a next week appointment or wait in the queue 2-3h and see my doctor on the same day. You can go for commercial insurance (which they actually call private) and cut these times.
Now, living in England and using the (fully public) NHS, I can't even get a GP appointment in the next 4 weeks. And the surgery mostly gives you only telephone appointment.
And no, you cannot just walk in and wait to see a doctor. Appointments only. This is not even only my surgery's problem.
The waiting time for specialist doctors are even more ridiculous.